Blow-up of Solutions to a $p$-Laplace Equation
Yuliya Gorb, Alexei Novikov · Multiscale Modeling and Simulation · 2012
Consider two perfectly conducting spheres in a homogeneous medium where the current-electric field relation is the power law. Electric field $E$ blows up in the $L^\infty$-norm as $\delta$, the distance between the conductors, tends to zero. We give here a concise rigorous justification of the rate of this blow-up in terms of $\delta$. If the current-electric field relation is linear, see similar results obtained earlier in [E. S. Bao, Y. Y. Li, and B. Yin, Arch. Ration. Mech. Anal., 193 (2009), pp. 195--226; H. Kang, M. Lim, and K. H. Yun, preprint, http://arxiv.org/abs/1105.4328v1, 2011; M. Lim and K. Yun, Comm. Partial Differential Equations, 34 (2009), pp. 1287--1315; K. Yun, SIAM J. Appl. Math., 67 (2007), pp. 714--730; K. Yun, J. Math. Anal. Appl., 350 (2009), pp. 306--312].